The behavior of complex fluids when being inside a porous medium may be very different from that when they are not. This is a problem encountered in many biological or industrial applications ranging from impregnation of fibrous materials to immiscible multi-phase flow in porous media. Among the different types of non-Newtonian fluids, many undergo behavioral changes depending on the stress or strain applied.
One can mention the Carreau rheology which is Newtonian at low shear rate but behaves as a power law fluid above a certain shear rate
[1]}. Other examples are yield stress fluid that responds like a solid below a critical yield threshold. Above, the materials
behave like a power law fluid [2]}. At the mesoscopic level, this rheological approach can also be extended to other situations.
For example, inertial effects can be described as a rheological change from a Newtonian fluid to a power law (quadratic or cubic) for a certain large Reynolds number [3]}. Another possible extension is the displacement immiscible fluids in porous media.
In this case, the fluids may each be Newtonian. However, the interfacial tension between them,
makes them effectively behave in a non-Newtonian way inside the porous medium [4]}.
Indeed, a minimum amount of stress is then required for a non-wetting phase to invade small pore throats.
